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Abstract

We give quantitative signed certificates retaining every arithmetic shift and mixed term of a weighted Weil form, and construct exact null corrections in its original norm. For each fixed ε>0, the specified inverse-receiving construction admits a trial cutoff m_M≤M^(1+ε) for all sufficiently large M. In the guarded input format, construction and verification have polynomial bit cost in the numeric cutoff M. The polar constraint is retained. Under RH, the admitted fixed instances have recognizable strict certificates. We also prove that the closed span of the canonical derivative generators equals the full-multiplicity Mellin zero-jet class in the original reciprocal-theta norm. An equivalent continuous-area norm and uniform bounds on zero-free exterior arms yield strong inward approximation. Contour moments give a computable orthogonal projection and optimal exact lifts of any fixed controllable finite head. At exact unit row moment and profile tolerance η, the logarithm of the least genuine-null physical norm has order η^(-4/(8j+13)) for a fixed row of highest index j. Simultaneous lifts have physical squared-norm budget exp(D_J e^(2L)), with D_J=exp(C(J+1)) and L≥C(J+1) for fixed auxiliary parameters, and allow prescribed orders J=o(log log M). A complete evaluated certificate and paid localization give the whole-domain lower bound −3×10^(-8) in the original physical norm. The covariance estimates and critical signed criteria connect these results to arithmetic Weil positivity.